3.535 \(\int \sqrt{x} (2+b x)^{3/2} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (b x+2)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]

[Out]

(Sqrt[x]*Sqrt[2 + b*x])/(2*b) + (x^(3/2)*Sqrt[2 + b*x])/2 + (x^(3/2)*(2 + b*x)^(
3/2))/3 - ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]]/b^(3/2)

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Rubi [A]  time = 0.0533578, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}+\frac{1}{3} x^{3/2} (b x+2)^{3/2}+\frac{1}{2} x^{3/2} \sqrt{b x+2}+\frac{\sqrt{x} \sqrt{b x+2}}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(2 + b*x)^(3/2),x]

[Out]

(Sqrt[x]*Sqrt[2 + b*x])/(2*b) + (x^(3/2)*Sqrt[2 + b*x])/2 + (x^(3/2)*(2 + b*x)^(
3/2))/3 - ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]]/b^(3/2)

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Rubi in Sympy [A]  time = 8.75426, size = 73, normalized size = 0.89 \[ \frac{\sqrt{x} \left (b x + 2\right )^{\frac{5}{2}}}{3 b} - \frac{\sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{6 b} - \frac{\sqrt{x} \sqrt{b x + 2}}{2 b} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+2)**(3/2)*x**(1/2),x)

[Out]

sqrt(x)*(b*x + 2)**(5/2)/(3*b) - sqrt(x)*(b*x + 2)**(3/2)/(6*b) - sqrt(x)*sqrt(b
*x + 2)/(2*b) - asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(3/2)

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Mathematica [A]  time = 0.0633419, size = 60, normalized size = 0.73 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (2 b^2 x^2+7 b x+3\right )}{6 b}-\frac{\sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(2 + b*x)^(3/2),x]

[Out]

(Sqrt[x]*Sqrt[2 + b*x]*(3 + 7*b*x + 2*b^2*x^2))/(6*b) - ArcSinh[(Sqrt[b]*Sqrt[x]
)/Sqrt[2]]/b^(3/2)

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Maple [A]  time = 0.007, size = 87, normalized size = 1.1 \[{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}}+{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{bx+2}}+{\frac{1}{2\,b}\sqrt{x}\sqrt{bx+2}}-{\frac{1}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(3/2)*x^(1/2),x)

[Out]

1/3*x^(3/2)*(b*x+2)^(3/2)+1/2*x^(3/2)*(b*x+2)^(1/2)+1/2*x^(1/2)*(b*x+2)^(1/2)/b-
1/2/b^(3/2)*(x*(b*x+2))^(1/2)/(b*x+2)^(1/2)/x^(1/2)*ln((b*x+1)/b^(1/2)+(b*x^2+2*
x)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + 2)^(3/2)*sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219929, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b^{2} x^{2} + 7 \, b x + 3\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 3 \, \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{6 \, b^{\frac{3}{2}}}, \frac{{\left (2 \, b^{2} x^{2} + 7 \, b x + 3\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} - 6 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{6 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + 2)^(3/2)*sqrt(x),x, algorithm="fricas")

[Out]

[1/6*((2*b^2*x^2 + 7*b*x + 3)*sqrt(b*x + 2)*sqrt(b)*sqrt(x) + 3*log(-sqrt(b*x +
2)*b*sqrt(x) + (b*x + 1)*sqrt(b)))/b^(3/2), 1/6*((2*b^2*x^2 + 7*b*x + 3)*sqrt(b*
x + 2)*sqrt(-b)*sqrt(x) - 6*arctan(sqrt(b*x + 2)*sqrt(-b)/(b*sqrt(x))))/(sqrt(-b
)*b)]

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Sympy [A]  time = 21.919, size = 92, normalized size = 1.12 \[ \frac{b^{2} x^{\frac{7}{2}}}{3 \sqrt{b x + 2}} + \frac{11 b x^{\frac{5}{2}}}{6 \sqrt{b x + 2}} + \frac{17 x^{\frac{3}{2}}}{6 \sqrt{b x + 2}} + \frac{\sqrt{x}}{b \sqrt{b x + 2}} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+2)**(3/2)*x**(1/2),x)

[Out]

b**2*x**(7/2)/(3*sqrt(b*x + 2)) + 11*b*x**(5/2)/(6*sqrt(b*x + 2)) + 17*x**(3/2)/
(6*sqrt(b*x + 2)) + sqrt(x)/(b*sqrt(b*x + 2)) - asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)
/b**(3/2)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + 2)^(3/2)*sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError